Numerical investigation of gas explosion phenomena in confined and obstructed channels

DOCTORAT DE L'UNIVERSITÉ DE TOULOUSE

Délivré par :

Institut National Polytechnique de Toulouse (Toulouse INP) Discipline ou spécialité :

Energétique et Transferts

Présentée et soutenue par :

M. OMAR DOUNIA le lundi 23 avril 2018

Titre :

Unité de recherche : Ecole doctorale :

Etudes des phénomènes d'accélération de flammes, transition à la

détonation et d'inhibition de flammes

Mécanique, Energétique, Génie civil, Procédés (MEGeP)

Centre Européen de Recherche et Formation Avancées en Calcul Scientifique (CERFACS) Directeur(s) de Thèse :

M. THIERRY POINSOT M. OLIVIER VERMOREL

Rapporteurs :

M. ASHWIN CHINNAYYA, ENSMA POITIERS M. DENIS VEYNANTE, ECOLE CENTRALE PARIS

Membre(s) du jury :

M. PAUL CLAVIN, AIX-MARSEILLE UNIVERSITE, Président Mme NABIHA CHAUMEIX, CNRS ORLEANS, Membre

M. OLIVIER VERMOREL, CERFACS, Membre M. THIERRY POINSOT, CNRS TOULOUSE, Membre

A trois personnes qui me sont chères, Ma mère, mon père et mon frère

Je tiens à remercier les membres du jury d’avoir accepté d’évaluer ce travail de thèse. Les différentes remarques émanant des deux rapporteurs, Pr. Denis Veynante and Pr. Ashwin Chinnayya, m’ont été d’une grande aide pour préparer la soutenance de thèse et m’ont certainement permis de clarifier certains aspects de ce manuscrit. La session de questions qui a suivi la présentation était une expérience très enrichissante. Les questions, toujours intéressante, de Pr. Nabiha Chaumeix et les interventions, jamais dénuées de sens et toujours théâtrales de Pr. Paul Clavin, y sont certainement pour quelque chose. Cette thèse n’aurait certainement pas été la même sans les conseils de mes deux directeurs de thèse. Le lecteur pourra sentir l’empreinte de Thierry, avec son recul et son expérience, dans les différentes parties de ce manuscrit. Aussi, l’oeil aiguisé et le tact absent de Thierry vous rappellent très souvent qu’il est inutile d’essayer de tromper le chef. Je suis bien obligé aujourd’hui de t’avouer, pauvre thésard contraint de lire ce manuscrit, que j’ai bien essayé. Mes tentatives sont toutefois restées vaines. Ensuite, il y a Olivier que je ne remercierai jamais assez pour ses différents conseils et son impact monstre dans le travail présenté dans ce manuscrit. Parler d’Olivier est certainement la partie la plus difficile de ces remerciements. Comment résumer ce que je pense de ce grand Monsieur en quelques lignes? Est-ce seulement possible? Ce sont les questions qui me taraudent l’esprit à l’heure où j’écris ce paragraphe. Laissant ça pour une autre occasion, je me permets une simple révérence: Chapeau bas l’artiste!

Je souhaite aussi profiter de ces remerciements pour exprimer ma gratitude envers les autres séniors du CERFACS (tout particulièrement Anto), au service informatique (pour leur efficacité incroyable et leur disponibilité), au service administratif et au secrétariat (pour leur bonne humeur inconditionnelle).

Place maintenant aux copains! Ces dernières années au CERFACS ont été très agréables pour moi. C’était toujours un plaisir de me rendre au CERFACS. Et c’est en parti dû à l’ambiance qui y règne et aux personnages toujours plus loufoques que l’on peut y trouver. Je commence par Pierre, mon co-bureau pour un peu moins de trois ans. Je pense avoir fait un portrait assez fidèle de Pierre lors d’une fameuse formation (n’est-ce pas Pierre!). Une petite dédicace quand même à tes t-shirts palmiers, tes jeans troués, ta bonne humeur et nos soirées FIFA. Merci Champion. On ne peut bien évidemment pas parler de Pierre sans parler de Biolchi, ce personnage étrange mais sympathique, fou mais attachant. Je tiens par la présente à adresser mes hommages à ses parents pour l’avoir mis au monde et mes félicitations à toute sa famille et ses proches pour avoir réussi à le côtoyer. Ce n’était certainement pas facile et il faut leur reconnaître ce courage. Je tiens aussi à m’excuser pour ce fameux incident de la verveine.

Travailler au CERFACS, c’est aussi côtoyer certaines crapules. Dans ce monde de crapules, Felix est certainement le roi. Je ne doute pas une seconde qu’il réussira dans la vie (surement en écrasant d’autres personnes dans son chemin, mais ça ça ne semble pas forcément le déranger). Dans le même genre, il y a aussi Nico Iafrate. J’émet par contre des réserves sur son avenir! Le bougre est bien capable de faire une crasse déplacée à son chef pour son premier jour de travail.

Je tiens aussi à remercier Valou, grâce à qui mes horaires de bureau sont devenus plus stables. Cette thèse aurait pris plus de temps à se finir sans toi ma poule. Merci!

mot pour lui, un sacré personnage), DaMel (à la base deux personnes Dario et mélissa, ils ont fini par fusionner vers la moitié de ma thèse. Apparemment, ils ne sont toujours pas ensemble.) etc...

Je termine ces remerciements par une petite citation:

Un homme sérieux a peu d’idées. Un homme à idées n’est jamais sérieux.

3

I

General concepts of self-propagating combustion

waves

1 Introduction

5

1.1 Explosion hazards . . . . 5

1.2 Direct detonation initiation . . . . 7

1.3 Delayed detonation initiation . . . . 8

1.4 Including DDT in risk assessment: past accidents . . . . 11

1.5 From preventive to mitigative measures: flame inhibition . . . . 11

1.6 The role of Safety Computational Fluid Dynamics (SCFD) . . . . 13

1.7 Objective of the thesis . . . . 14

2 Conservation equations and models for reactive flows 17 2.1 Introduction . . . . 17

2.2 Multi-species reactive flows . . . . 17

2.3 Compressible multi-species reactive Navier-Stokes equations . . . . 18

2.4 Different approaches to kinetic modeling . . . 20

2.5 Computational approaches for reactive Navier-Stokes equations . . . . 22

2.6 Large Eddy Simulations (LES) . . . . 24

2.7 Numerical aspects . . . . 26

3 Theoretical concepts of premixed flames 29 3.1 Introduction . . . . 29

3.2 Laminar premixed flames . . . . 29

3.3 Turbulent premixed flames . . . . 39

4.2 One-dimensional structure of detonations . . . . 47

4.3 Selection mechanism of CJ detonations . . . . 51

4.4 Instabilities of detonation fronts . . . . 53

5 Computation strategies for gas explosion simulations 57 5.1 Flame acceleration . . . . 57

5.2 DDT . . . . 57

5.3 Inhibited FA . . . . 58

60

II

What are the mechanisms responsible for the extreme

damages

observed during VCE’s?

6 The Gravent explosion channel 63 6.1 Experimental procedure . . . . 64

6.2 Influence of the mixture composition on the explosion scenario . . . . 64

7 LES of flame acceleration in a confined and obstructed channel 67 7.1 Numerical setup . . . . 67

7.2 Validation of the LES . . . . 70

7.3 Influence of the thermal boundary condition . . . . 74

7.4 Flame acceleration mechanisms involved in the BR30hS300 channel . . . . 76

8 The role of shocks in the spontaneous initiation of detonations 93 8.1 Spontaneous initiation of detonations . . . . 93

8.2 On the difficult process of generating hot spots . . . . 96

8.3 The capacity of strong shocks to trigger DDT . . . . 99

8.4 No universal mechanism . . . 102

9 Numerical investigation of the DDT mechanisms in the stoichiometric case 105 9.1 Focus on DDT . . . 105

9.2 The detonation front . . . 141

10 Mitigation of vapour cloud explosions 147

10.1 Physical mitigation using water sprays . . . 148 10.2 Dominantly chemical mitigation using metal containing compounds . . . . 149 10.3 The mechanism of flame inhibition by metal salts . . . 150 10.4 On the conditions required for VCE mitigation by sodium bicarbonate powder153 10.5 Objective of the present study . . . 155

11 Computational approach to flame inhibition by sodium bicarbonate 156

11.1 Lagrangian formalism for the solid phase . . . 156 11.2 Gas-phase inhibitor/flame interaction . . . 158 11.3 The TFLES model for multi-step chemistries . . . 162

12 Investigation of the mechanism of flame inhibition by sodium

bicarbon-ate particles 164

12.1 Influence of sodium bicarbonate powders on self-propagating flames . . . . 164 12.2 Annexe: Deflagration inhibition by sodium bicarbonate: a demonstration

case . . . 204

208 Conclusions

Part I

General concepts of self-propagating

combustion waves

Introduction

1.1

Explosion hazards

Every year, mining, process and energy industries suffer billions of dollars of losses world-wide due to gas explosions. In addition to the direct damage to the industrial plants, these explosions often lead to long business interruptions, causing secondary costs that need to be considered. Indeed present-days production chains exhibit a high degree of consolidation, which, in case of explosion, can lead to catastrophic financial losses. More-over, explosion accidents are often tragic and lead to a high number of severe injuries and fatalities. Many severe industrial disasters have been reported in the past years:

• April 14th, 1944, India - Cargo explosion in Mumbai : 1300 fatalities, 3000 injuries; • July 11th, 1978, Spain - Tanker road accident: 217 fatalities, 200 injuries;

• November 19th, 1984, Mexico - Series of massive explosions in a liquid petroleum gas tank in San Juanico: 500 fatalities, 6000 injuries;

• March 23rd, 2005, US - Explosions in a BP Texas refinery: 15 fatalities, 180 injuries; • December 11th, 2005, England - Buncefield disaster: 43 injuries;

• April 20th, 2010, Golf of Mexico - Explosion in Off-shore petroleum platform: 11 fatalities;

• August 25th, 2012, Venezuela - Explosion in gas refinery in Amuay: 48 fatalities; • August 12th, 2015, China - Series of massive explosions at a container storage station

in the Port of Tianjin: 173 fatalities, 797 injuries.

This non exhaustive list of disasters shows that large explosion hazards can have un-foreseeable social and political impacts that can exceed the direct economical damages. Sound knowledge of explosions physics is therefore of vital importance for the prediction of these extreme scenarios.

First, the term “explosion” needs to be defined. It refers to a violent event able to gen-erate considerable overpressure. An insightful definition was proposed by F. M. Global

(2013): “an explosion is a rapid transformation of potential physical or chemical energy into mechanical energy and involves the violent expansion of gases”. In this definition, the

Figure 1.1: Position of the Vapour Cloud Explosions (VCE) in the classification of explosions

in chemical process industries proposed by Abbasi et al.(2010).

terms “physical” and “chemical” refer to the way the energy, that enabled the explosion, is accumulated:

• Physical explosions are caused by the sudden release of an energy accumulated by gas or volatile liquid heating in a containment for example, or by simple gas compression, generating overpressure. The release can be the result of the failure of the containment.

• Chemical explosions are due to the release of energy by chemical reactions causing pressure build-up. Chemical explosions can be either homogeneous or reaction front driven.

Independently of their nature, explosions generate overpressure. The level of pressure pro-duced during an explosion is the critical parameter that will determine the severity of the damages inflicted on buildings and civilians. Because the losses vary considerably from an explosion to another, it seems that some mechanisms can greatly increase the levels of overpressure generated. Identifying and understanding these mechanisms is crucial to provide engineers with guidelines for implementing preventive and mitigative measures.

Abbasi et al. (2010) proposed different subclassifications of explosions in process in-dustries. A part of this classification is shown in Fig. 1.1. The mechanisms involved vary considerably from an explosion to another. This thesis focuses on reaction front driven explosions and more specifically on Vapour Cloud Explosions (VCE). A sequence of events commonly involved in VCEs is displayed in Fig. 1.2. Reactive explosions occur when a cloud of premixed reactive mixture has enough time to form after a gas leak. An external source of energy Es deposited over a duration τs can then ignite the mixture

and a reaction wave emerges from the source point, consuming the reactants ahead of it to form products at its tail. The origin of this external energy is generally attributed to sparks, walls heating, annex explosions etc... Generally speaking, two distinct types of self-propagating reaction waves can emerge from the region where energy is deposited (their structure is shown in Fig. 1.3):

Figure 1.2: Commonly observed event tree leading to gas explosions in buildings.

propagation is governed by thermal conduction as well as by chemical reaction. The temperature inside the flame front gradually increases due to, first, diffusive flux of energy by conduction, before exothermic reactions increase the temperature further to reach the equilibrium temperature of the products. Premixed flames are initiated if Es is relatively low but sufficient to activate chemical reactions.

The term “deflagration” generally refers to premixed flames that have reached high propagation velocity with respect to a fixed point.

• Detonations are supersonic waves. Their propagation is governed by shock dynam-ics. Contrary to premixed flames, the reaction zone is not initiated by heat diffusion. Instead a strong shock brings the reactants to the Von-Neumann state (VN) where the temperature is high enough to initiate reactions. Detonations can be directly initiated if Es is sufficiently high and deposited on a very low time scale τs.

How-ever, this is not a necessary condition and detonations can be triggered by another mechanism involving the spontaneous ignition of a pocket of fresh gases provided some precondition is met.

Because of the presence of a strong shock coupled to a reaction zone, detonations are fare more destructive than deflagrations. Therefore, the initiation of the former waves has to be carefully studied to identify the possible mechanisms that can trigger this escalation in the explosion hazard. Generally speaking, detonations are observed when: 1) a strong blast forms and decays to a self-propagating detonation front; 2) a premixed flame forms and creates, during its propagation, the ideal conditions for the coupling between a shock and a reaction wave, thereby giving rise to a detonation. Both scenarios are discussed briefly in Sections 1.2 and 1.3.

1.2

Direct detonation initiation

A detonation wave can be initiated far away from the energy deposition locus if sufficient energy Es is deposited (quasi-)instantaneously (Clavin and Searby (2016); Lee (2008)).

Figure 1.3: The inner structure of premixed flames and detonations. In a premixed flame

(left), reactions are initiated by the gradual increase of temperature due to the diffusive flux of heat. The reaction zone, in a detonation (right), is initiated by the abrupt temperature increase induced by the leading shock. The indices u and b indicate the fresh and burnt gases respectively. The bold arrow indicates the direction of propagation.

Es must then be higher than a certain critical value Es,1c (see Fig. 1.4(a)). In this case, a

strong blast wave is formed and decays to a self-propagating detonation wave at a certain distance rd from the ignition source (see Fig. 1.4(b)). Once the detonation regime is

reached, a quasi-steady high overpressure level is observed. Note that rd depends on Es.

On the other hand, no detonation is observed if Es is just sufficiently high to initiate a

blast wave (i.e. Es> Es,2c ) but lower than the critical value Es,1c . The blast simply decays

to a relatively weak shock. Note that the damages inflicted on buildings and civilians can be extreme in these cases because of the shock generated initially.

In terms of safety, it is important to predict the critical energy deposition Ec

s,1that can lead

to detonation onset. This has been assessed both theoretically (He and Clavin(1994)) and experimentally (Bach et al.(1969);Lee(1984)). E_s,1c was found to be quite large and can explain the formation of gaseous detonations in free space. In this thesis, we are interested in semi-confined and confined configurations, in which direct detonation initiation is not the commonly observed scenario. There is, however, a mechanism by which a detonation can still form far away from a mild ignition source if certain conditions are met. This is discussed briefly in the next section.

1.3

Delayed detonation initiation

This thesis focuses on explosions occurring on configurations that are either semi-confined or confined. In such configurations, the explosion is generally initiated by a mild ignition, i.e. Es  Es,2c . A deflagration front then emerges from the ignition source. Since a

Figure 1.4: (a) Depending on the energy deposited Esto ignite the mixture, multiple scenarios can arise. This thesis addresses the case of a mild ignition that leads to deflagration initiation. (b) Typical maximum overpressure (overP) evolution with the distance r_s from the ignition source. When the energy input to the system is sufficient, i.e. Es> Ecs,1, a blast wave transitions to a detonation wave with a quasi-steady high overpressure level. Conversely, if E_s,2c < Es< Es,1c , the blast wave decays to form a relatively weak shock.

acceleration process that leads to the fast flames observed during explosions. A “strong” acceleration stage is a necessary condition before a detonation initiation can be observed.

1.3.1

Flame acceleration: from very subsonic to very fast flames

As discussed above, a VCE can be defined as a “rapid” release of chemical energy. Given that premixed flames are subsonic combustion waves, one may ask where the term “rapid” comes from, in the case of deflagrations. The answer lies in the ingredients controlling the flame propagation speed: 1) flames consume the reactants at their vicinity at velocity Vconsumption that depends on the local conditions in the fresh gases (Vconsumption is also

the velocity of the flame with respect to the fresh gases); 2) similarly to acoustic waves, flames are also advected by the flow at speed Vf low. With respect to a fixed point, the

flame propagation speed Vf lameis then given by the relation: Vf lame = Vconsumption+ Vf low.

An important feature of self-propagating flames is that Vconsumption can be influenced by

numerous mechanisms, described in Chapter 3, which makes these reaction waves intrin-sically unstable (Lee (2008)). A constructive feedback between the unsteady flame and the flow ahead of it may give rise to a Flame Acceleration (FA) process responsible for the emergence of the fast flames observed in real explosions.

As the flame accelerates, the exothermal reactions inside the deflagration front release more energy leading to more pressure build-up. A typical FA scenario is displayed in Fig. 1.5(a). The maximum overpressure, and consequently the degree of damages ob-served during the gas explosion, depends on the final level of acceleration reached by the deflagration and varies considerably from a configuration to another.

In order to predict the maximum overpressure reached during FA in VCEs, empirical methods are still used nowadays to design industrial plants. A compilation of such for-mulas is proposed in Bradley and Mitcheson (1978) and Molkov et al. (1997). However, their applicability is limited to cases identical (or very close) to the setup on which such

Figure 1.5: (a) The solid line represents a typical flame acceleration scenario that leads to

high damages. The bold line corresponds to the case where DDT is observed with a drastic increase in the damages inflicted. (b) A deviation from the typical FA scenario is observed if mitigative procedures are applied. A good mitigative procedure for VCE does not result in flame extinction, instead the flame must continue to burn reactive gases but with minimum damage.

correlations have been built. Indeed, FA is controlled by instabilities and as such is highly sensitive to initial and boundary conditions (in addition to other parameters discussed later in the thesis). Such correlations can hardly take into account the complexity of the industrial structures and a deviation of the observed overpressure from the one predicted with such correlations is often observed.

1.3.2

Deflagration to detonation transition

Flame acceleration results in very fast deflagrations, sometimes supersonic with respect to the laboratory frame. Once this regime is reached, the appropriate conditions for det-onation onset may be reached as displayed in Fig. 1.5(a). This transition is called the Deflagration to Detonation Transition (DDT). A variety of events/ingredients have been shown, theoretically, experimentally and numerically, to trigger DDT under specific flow conditions. The challenge is to understand the interplay among these ingredients and the relative importance of each one.

DDT marks a brutal escalation in the explosion hazard. To implement preventive mea-sures against DDT, multiple criteria, summarized in Ciccarelli and Dorofeev(2008), have been proposed. Assuming that a deflagration has already accelerated to a certain critical supersonic velocity that depends on the configuration, it is possible to define a criterion for the onset of detonation. The problem is that boundary conditions play an important role in both the onset and the propagation of detonations, so that it is only meaningful to discuss possibility of DDT for specific boundary conditions (see Lee(2008) and Chap-ter 8 of the manuscript). Therefore, such criteria can hardly be of relevance to complex industrial plants.

1.4

Including DDT in risk assessment: past accidents

Large scale experiments, mentioned in Ciccarelli and Dorofeev (2008) and Lee (2008), provide ample evidence that DDT can occur in a VCE. Not only that, the detonation, once initiated, can continue through unobstructed regions of the cloud, thereby inflicting damage to areas far away from the onset locus. The experimental evidence points out the necessity to include the DDT phenomenon in risk assessment, despite its low probability of occurence. Recent industrial accidents show that a simple tank switch failure can trigger a cascade of events leading to huge industrial disasters:

• Buncefield, UK, 2005: a tank, filled with unleaded petrol, began to overflow. This was attributed to the failure of a switch, which would have detected that the tank was full and shut off the supply. This resulted in the rapid formation of a rich fuel and air vapour. By the time the first explosion occurred, the cloud had spread beyond the boundaries of the site. Therefore, the devastated area exceeded the industrial site. The investigation report, issued by the british Government in 2008, pointed out some indicators of DDT. In particular, the damages recorded on areas that do not present signs of congestion (see Fig. 1.6 (top)) suggest the self-propagation of a detonation front. The disaster did not result in any fatality but it caused over 200 million euros of financial losses.

• Jaipur, India, 2009: a vapour petrol leak was detected but the non-observance of (normal) safety procedures led to a strong explosion. The evidence of severe pressure damage can be observed throughout the area, as illustrated in Fig. 1.6

(bottom). The level of damage observed indicates pressures of several bars and is consistent with the damage caused by the passage of a detonation. 11 people were killed in the disaster and the financial losses add up to nearly 40 million euros. These severe accidents demonstrate the necessity to implement measures than can either prevent or reduce the impact of explosions. In the last case, this procedure is called mitigative.

1.5

From preventive to mitigative measures: flame

inhibition

For safety engineers, there are two possible ways to deal with VCE’s:

• Preventive measures. The problem with these measures is that their effectiveness can hardly be always guaranteed. First, a large spectrum of mechanisms can lead to strong FA. Preventive measures may limit the impact of some mechanisms that are dominant under certain conditions. But they can fail when some unexpected scenarios give rise to other processes that can trigger a cascade of dramatic events as often highlighted in industrial disasters. Second, even though criteria exist for the detonation propagation limit beyond which the self-sustained propagation of a detonation wave is not possible, they are strongly dependent on the boundary conditions. Therefore, one can even question the universality of such preventive

Figure 1.6: Damages recorded during the Buncefield (top) and the Jaipur (bottom) incidents.

There are some indicators of DDT during these disasters.

measures when taking into account the complexity of the explosion scenario in real industrial plants.

• Mitigative measures. Since the first explosion accidents, it has become clear that preventive guidelines must be supported by mitigative procedures that can activate when a gas leak or an ignition is detected. Water sprays are an example of such measures. Their applicability to vapor cloud explosions has been shown by a number of authors (Acton et al. (1991); Bjerketvedt et al. (1997); Van Wingerden (2000)). Water droplets act as a heat sink and their effectiveness depends on their size. Water sprays are a promising mean for explosion mitigation since they are relatively cheap and most industrial installations are already equipped with a fire fighting system via which water can be delivered. However, there are also a number of drawbacks. The main one, identified by practical studies performed on the petrochemical complex of Samsung-Total Petrochemicals in South Korea (Hoorelbeke (2011)), is that the quantities of water needed to strongly reduce the overpressure may be huge.

For the case of VCE’s however, there is a condition that a mitigative measure must verify before it can be considered as an operational system: the cloud of reactive mixture that have formed due to the gas leak must be eliminated. This means that any procedure that can eventually result in flame extinction can not be applied to VCE’s, as highlighted in Fig. 1.5(b). Indeed, if at some point the flame is suppressed, a second explosion can still occur later in the remaining reactive cloud. Because of the high temperatures and pressures created by the first explosion, the second ignition may lead to higher damage. Therefore, an effective mitigative measure for VCE’s is one that strongly reduces the overpressure without extinguishing the flame. Because VCE’s are reaction front driven, an efficient way to achieve this is to reduce the overall exothermic reaction rate in the flame, so that only very weak FA is achieved during the deflagration propagation. This mechanism is referred to, by Linteris (2004), as flame inhibition and will be discussed in more details in PartIIIof the manuscript. It can be achieved, for example, by injection

of a powder ahead of the flame, as studied in this PhD.

1.6

The role of Safety Computational Fluid

Dynam-ics (SCFD)

As explained in the previous section, VCE scenarios are complex and controlled by var-ious mechanisms. The interplay among them is not yet entirely understood. The same remark applies to mitigative measures. The mechanism by which the latter act on the explosion has to be characterized in details, in order to avoid surprising counter-effects. Understanding all these intricate processes is of vital importance and requires detailed experimental diagnostics. Coupling accurate numerical simulations to well documented experiments can allow an elaborate description of these phenomena.

Considered too expensive a dozen years ago, CFD is now more and more included in risk assessment studies. The main advantage of CFD, compared to phenomenological and empirical methods, is to allow access to flow and flame structures. This is crucial to understand the mechanisms involved in VCE. The main limit to the use of SCFD is the complexity of the configurations considered. Industrial plants are often complex, includ-ing large scale (buildinclud-ings, tanks, etc...) to small scale (pipes, instrumentation devices, etc. ) structures. The same remark holds for the time scales (chemistry, instabilities, detonation, flames, shocks, etc.). The spectrum of time and length scales is large and implies huge computational resources, that may not even be available.

In order to reduce the computational cost of such simulations, modeling the entire tur-bulent spectrum is a usual approach known as Unsteady Reynolds Average Navier-Stokes (URANS). It has been adopted by most of the licensed SCFD codes. A series of bench-mark tests have been conducted Baraldi et al.(2009);Garcia et al.(2010);Makarov et al.

(2010) showing acceptable agreement with experimental data. It is however important to note that most of these simulations have been conducted with an a priori knowledge of the experimental results, allowing the user to tune the model coefficients to match the desired curves. According to Pope (2001), the most widely used RANS turbulent models (k − ε Jones and Launder (1972); Launder and Sharma (1974), k − ω Wilcox (2008)) can fail profoundly when applied to inhomogeneous turbulent flows. Even in the case of Reynolds-stress models (Pope (2001)) a calibration of the constants is necessary for many flows. The relevance of such models to transient phenomena like VCE is therefore doubtful. As indicated by Ciccarelli and Dorofeev (2008), “RANS codes are missing the essential physics that are responsible for flame acceleration and DDT and therefore can only be used for qualitative information”. Blind simulations of realistic explosion scenar-ios using these codes are still to be performed to assess their capacity to correctly predict the experimental observations.

With the increase of the computational power of present-days computers, the Large Eddy Simulation (LES) approach emerges as a promising alternative to URANS. In LES, the large flow structures are resolved explicitly whereas small structures are modeled. Many authors (Makarov and Molkov (2004); Molkov et al. (2007); Gubba et al. (2008, 2009);

Zbikowski et al. (2008); Ibrahim et al. (2009); Gubba et al. (2011); Wen et al. (2012);

Quillatre et al. (2013);Xu et al. (2015); Zhao et al. (2017);Tolias et al. (2017);Volpiani et al. (2017a); Vermorel et al. (2017)) have shown the capacity of LES to capture the transient effects related to turbulence and flame-turbulence interactions during

deflagra-Figure 1.7: Typical vapour cloud explosion scenarios addressed in this thesis.

tions in small-scale closed and vented chambers. However, LES shows serious limitations when used to compute deflagrations in large scale configurations (550m3 _{in the study of} Molkov and Makarov (2006)). In such domains, the grid resolution is degraded to keep the computational cost reasonable, which reduces considerably the part of the physics resolved. One may even question the LES designation in these cases.

While small scale deflagrations have been computed with good accuracy using LES, DDT has mostly been simulated with Direct Numerical Simulations (DNS) (see the work of Oran and coworkers summarized in Oran and Gamezo (2007)). The reason behind this is that very small grid cells are needed to capture both the flow, the flame and the det-onation structures. This reduces considerably the dimensions of the problem that can be considered. Even for “small” domains, three dimensional DNS is still unaffordable nowadays and in most cases only two-dimensional domains are considered. The 2D-DNS aims at highlighting important mechanisms that can trigger DDT.

1.7

Objective of the thesis

The objective of this thesis is to investigate numerically reaction front driven explosions. First, conservation equations for reactive flows and some modeling aspects are recalled in Chapter 2. Some theoretical aspects of premixed flames and detonations are discussed in Chapters 3and4respectively. FA, DDT and flame inhibition are investigated in this the-sis, each problem requiring a specific computational strategy. For the sake of clarity, the different methods considered are summarized in Chapter 5and a justification is provided whenever a change in strategy was deemed necessary. Parts II and III of the manuscript are dedicated to the presentation of the results.

As displayed in Fig. 1.4, all the scenarios considered in this work start with a mild igni-tion, from which a subsonic deflagration front emerges. This manuscript tries to answer

two major questions: 1) What are the mechanisms responsible for the extreme damages observed during VCE’s? 2) What is the underlying mechanism behind some efficient mitigative measures?

Part II: What are the mechanisms responsible for the extreme damages ob-served during VCE’s?

To answer this question, the scenario leading to DDT is investigated in two steps: a case resulting in fast deflagrations followed by a case where the transition is observed. The objective is to identify the necessary conditions that the flame acceleration phase must satisfy to trigger DDT. As discussed earlier, boundary conditions play an important role during both FA and DDT. Therefore, the discussion can only be meaningful if the same configuration is used for both cases. The Gravent explosion channel (Vollmer et al.(2012)) offers a large database of experimental results tailored for this investigation. The selected experiment, presented in details in Chapter 6, considers a long obstructed channel filled with premixed hydrogen/air mixtures at various compositions. Both FA and DDT have been observed depending on the equivalence ratio φ of the mixture.

Part II.a: Flame acceleration, from subsonic to very fast deflagrations

First a case corresponding to a lean (φ = 0.52) premixed hydrogen/air mixture is con-sidered. The results are presented in Chapter 7. The crucial importance of repeated flame-obstacle interactions in producing very fast deflagrations is highlighted. The high levels of pressure recorded during the experiments are explained. The study ends with a theoretical study of the flame journey in a flow contraction to deepen our understanding of the flame/obstacle interaction.

Part II.b: Transition from deflagration to detonation phenomena

By increasing the equivalence ratio of the mixture from φ = 0.52 to φ = 1, DDT is observed in the experiments. To identify the possible mechanisms that can explain DDT in this case, it is mandatory to understand why no detonation onset occurred in the case φ = 0.52 despite the strong shocks produced ahead of the flame (Chapter 8). This prepares the ground for the numerical investigation of the detonating case, presented in Chapter 9. Particular attention is drawn to the impact of the chemistry modeling on the detonation scenario.

Part III. What is the underlying mitigative mechanism behind some solid in-hibitors?

Part II focuses on possible VCE scenarios when no mitigative measure is applied. As discussed in Section 1.5, it is possible to considerably reduce the pressure levels produced during these events using techniques that are suited for reactive front driven explosions. Chapter 10 introduces the general concept of the chemical flame inhibition and its appli-cability to gas explosions. A particular attention is drawn to the sodium bicarbonate, a very efficient solid inhibitor. Chapter 11 presents the different methods used to derive a model able to reproduce the first order effects of sodium bicarbonate particles on a self propagating methane/air flame with reasonable computational cost. This model is then employed to investigate numerically the flame/inhibitor powder interaction. The results are discussed in Chapter12. First, Section12.1 focuses on the impact of the particle size

and powder stratification on the flame/inhibition process. Then, in Section 12.2, three-dimensional simulations of a methane/air deflagration inhibition by sodium bicarbonate powders are presented. The suitability of the model for three-dimensional simulation is demonstrated.

Conservation equations and models for

reactive flows

2.1

Introduction

The objective of the thesis is to investigate different gas explosion scenarios using numer-ical simulations. To describe fluids, a set of equations is needed. The first proposition of such a system of equations describing inviscid three-dimensional flows is attributed to Leonhard Euler. The notion of viscosity was later included by Louis Navier and George Gabriel Stokes to form the present-days formulation. Navier-Stokes equations are a set of partial differential equations including highly non-linear terms which, in most cases, make finding analytical solutions almost insurmountable. The very few analytical solutions have been obtained in general by considering cases where the non-linear terms vanish natu-rally. In most cases however, Navier-Stokes equations must be dealt with numerically. This chapter introduces the different strategies employed to “resolve” these equations.

2.2

Multi-species reactive flows

Reactive flows involve multiple species reacting through a set of chemical reactions. To characterize a species k in a mixture of mass m, containing n moles and N species, the mass fractions Yk and the mole fraction Xk are used:

Yk= mk/m (2.1)

Xk= nk/n (2.2)

where mk (nk) is the mass (number of moles) of species k in a given volume and m

(n) is the total mass (number of moles) in this volume respectively. By definition,

N X k=1 Yk= N X k=1 Xk= 1 (2.3)

The mean molecular weight of the mixture M is defined using the molecular weight of each species Wk : 1/M = N X k=1 Yk/Wk (2.4)

The heat capacities at constant pressure Cp and constant volume Cv are defined by: Cp = N X k=1 YkCp,k (2.5) Cv = N X k=1 YkCv,k (2.6)

where Cp,kand Cv,k are respectively the heat capacities at constant pressure and constant

volume of species k.

2.3

Compressible multi-species reactive Navier-Stokes

equations

Navier-Stokes equations include conservation equations for mass, momentum and energy, plus a conservation equation for each species considered in the mixture. When radiative source terms are neglected, these equations have the following form:

∂ρ ∂t + ∂ ∂xj (ρuj) = 0 (2.7) ∂ ∂t(ρui) + ∂ ∂xj

(ρuiuj + pδi,j− τi,j) = 0 (2.8)

∂

∂t(ρE) + ∂ ∂xj

(ui(ρE + pδi,j− τi,j) + qj) = ˙Q + ˙ωT (2.9)

∂ ∂t(ρYk) + ∂ ∂xj (ρYkuj) = − ∂ ∂xj (Jj,k) + ˙ωk (2.10)

where the Einstein notational convention is used. ρ, (ui)i=1,3 and p are the density, the

velocity and the static pressure respectively. Q denotes an external energy source term˙

that can be induced by a spark or a laser for example. In this thesis, ρ and p are related through the state equation: ρ = rT /p, where T is the temperature and r = R/M is the perfect gas constant R = 8.3145J mol−1K−1 divided by the mean molecular weight M . E is the sum of the sensible and kinetic energy.

Viscous and pressure tensors: τ

i,j

and −pδ

i,j

In both momentum (2.8) and energy (2.9) conservation equations, the term σi,j = τi,j −

pδi,j appears as the sum of the viscous tensor τi,j and the pressure tensor −pδi,j. τi,j is

defined by: τi,j = − 2 3µ ∂uk ∂xk δi,j+ µ ∂ui ∂xj + ∂uj ∂xi ! (2.11) where µ is the dynamic viscosity. δi,j is the Kronecker symbol: δi,j = 1 if i = j and 0

Diffusive flux of species J

i,k

In the species conservation equation (2.10), the diffusive flux of species Ji,k appears in the

right hand side. The exact evaluation of Ji,k is complex since it involves species diffusion

velocities which are obtained by solving a linear system of size N2 _{in each direction of}

space, for each point and each time. Since mathematically this task can be difficult and costly (Ern and Giovangigli(1994)), an approximation is commonly used in most numeri-cal codes. TheJ. O. Hirschfelder and Bird(1954) approximation replaces the binary mass diffusion coefficient Dk,j of species k into species j by an equivalent diffusion coefficient

of species k into the rest of the mixture Dk. This first order approximation however does

not guarantee mass conservation and a correction velocity (Vc

i )i=1,3 is introduced: Ji,k = −ρ Dk Wk M ∂Xk ∂xi − YkVic ! (2.12) where V_ic = N X k=1 Dk Wk M ∂Xk ∂xi (2.13)

Energy flux q

i

For a multi-species flow, the energy flux qi is the sum of two terms: heat conduction and

diffusion of species with different enthalpies.

qi = −λ ∂T ∂xi − ρ N X k=1 Dk Wk M ∂Xk ∂xi − YkVic ! hs,k (2.14)

where hs,k is the sensible enthalpy of species k and λ is the heat diffusion coefficient.

Species source terms

Consider a chemical system of N species reacting through M reactions:

N X k=1 ν_kj0 Mk N X k=1 ν_kj00 Mk for j = 1, M (2.15)

where Mkis a symbol for a species k. νkj0 and ν

00

kj are the molar stoichiometric coefficients

of species k in reaction j. The mass reaction rate ˙ωk of species k is the sum of the rates

˙

ωkj produced by each reaction:

˙ ωk = M X j=1 ˙ ωkj = Wk M X j=1 νk,jQj (2.16)

where νk,j = νkj00 − νkj0 and Qj is the progress rate of reaction j which is defined by:

Qj = Kf,j N Y k=1 _ρY k Wk νkj0 − Kr,j N Y k=1 _ρY k Wk νkj00 (2.17)

Kf,j and Kr,j are the forward and reverse rates of reaction j and are usually modeled

using the empirical Arrhenius law:

Kf,j = Af jTβjexp (−Ea,j/RT ) (2.18)

The reverse rates Kr,j are computed from the forward rates through the equilibrium

constants: Kr,j = Kf,j _p a RT PN k=1νk,j exp  ∆S0 j R − ∆H0 j RT  (2.19)

where pa = 1bar. ∆Hj0 and ∆Sj0 are respectively the enthalpy (sensible and chemical)

and entropy changes for reaction j: ∆S_j0 = N X k=1 νk,jWksk(T ) (2.20) ∆H_j0 = N X k=1 νk,jWk(hs,k(T ) + ∆h0f,k) (2.21) ∆h0

f,k is the enthalpy of formation of species k at temperature T0 = 0K, and allows to

defined the heat release due to combustion ˙ωT:

˙ ωT = − N X k=1 ˙ ωk∆h0f,k (2.22)

Finally the sensible enthalpy hs,k =

RT

T0Cp,kdT is defined to satisfy hs,k(T = T0) = 0. The

sensible energy is then defined by: es,k =

RT

T0Cv,kdT so that es,k = hs,k− RT /Wk.

Transport properties

To evaluate transport properties at reasonable cost, other approximations are applied. First, the Prandtl number P r = µCp/λ, which compares the momentum and heat

trans-port, is set constant. Second, the Schmidt number Sck = µ/(ρDk) of each species k,

which compares momentum and species diffusion, is assumed constant as well. This leads to a simplified relation for the species diffusion coefficient, which can then be evaluated using the viscosity µ. In this thesis, the temperature dependence of µ is assumed to follow a power law:

µ = µref(T /Tref)b (2.23)

where µref is the viscosity at T = Tref, and b a constant of the model in the range [0.5, 1.]

2.4

Different approaches to kinetic modeling

The species chemical source terms ˙ωk in the energy and species balance equations (2.9,

2.10) describe the species conversion rates during the combustion process. “Real” flames involve many reactive molecules with a large range of reaction time scales τr. Thanks to

collision theory and quantum mechanics codes, the rate kr of chemical reactions can be

evaluated. In general, an Arrhenius law is used to get the expression of kr:

kr≈ A exp(−Ea/RT ) (2.24)

This law basically states that the energy brought by colliding reactive molecules has to be higher than a certain energy Ea to open bonds and initiate reaction. Furthermore, the

colliding molecules must be oriented in a manner favorable to the necessary rearrangement of atoms and electrons. The frequency of collisions in the correct orientation is given by A. To take into account the dependence of this collision frequency on temperature, A exhibits usually the form: A ≡ ATβ_{. It is important to note that the Arrhenius formulation is}

a drastic assumption of the intricate molecular collision phenomena. We present in the following a brief overview of different approaches to chemistry description that are based on an explicit chemical scheme. Tabulation methods (Maas and Pope (1992); Gicquel et al. (2000); van Oijen et al. (2001); Fiorina et al. (2003, 2010); Auzillon et al. (2011,

2012)), which are also commonly used in the combustion community, will not be described in the following.

Multiple approaches for chemistry description

Assembling a complete set of elementary reactions for a global combustion process re-sults in a detailed chemical kinetic mechanism. It is important to note that such chemical schemes are still considered as models (see S. (2003)) since that reaction rate expressions still rely heavily on intuition: 1) the Arrhenius formulation is an approxima-tion; 2) an optimization procedure is applied in all/some rates constants to match targets from the literature; 3) a large sensitivity of the results on some rates can be observed. These mechanisms are generally validated against experimental data for a large range of operating conditions and can therefore accurately reproduce a number of combustion processes. They are in practice used in chemical codes, like CANTERA Goodwin et al.

(2017), or some DNS codes Patnaik et al. (1989); Tanahashi et al. (2000); Arani et al.

(2017) mainly for hydrogen combustion because of the low number of species and reac-tions involved. Recently DNS of a lean methane/air turbulent premixed flame has been performed (Aspden et al. (2016)) using a detailed chemistry description. However, the use of detailed chemical schemes is still too expensive to be considered in most realistic configurations for most hydrocarbon fuels. The problem is the high number of reactions and species involved which increases drastically the computational cost of the simulation. It is possible to considerably reduce the number of species and reactions in these chemical schemes by removing the species and the reactions that are irrelevant to the problem considered. The validity of the skeletal mechanism obtained is then restricted to a certain range of operating conditions. It is also possible to replace the transport equation of some species with negligible production rate by an algebraic expression for their mass fraction. Because it is much less expensive to compute an analytical expression instead of solving a transport equation, this procedure yields an important reduction of the global cost of such mechanisms. The latter are called ARC for Analytically Reduced Chemistries and are employed in this thesis to perform flame inhibition simulations. The reduction procedure will be detailed in Part III. In the CFD community, ARC schemes have been used in cases where considering radical species is either mandatory or can considerably improve the simulation results (seeJones and Prasad(2010);Franzelli et al.(2013);Schulz et al. (2017); Jaravel et al. (2017)).

Figure 2.1: Validity span of different approaches to kinetic modeling.

Until recently, a common approach to combustion modeling for CFD was the use of

Glob-ally reduced chemistries (GRC), usuGlob-ally composed of a number of species lower than

10 and a number of reactions lower than 4 (Jones and Prasad (2011); Franzelli et al.

(2013)). The idea behind these mechanisms is to reproduce global parameters (flame speed sL or induction times during ignition for example) that are assumed to dominantly

control the physics during the simulation. They are relatively simple, can be tuned easily to match certain targets and are cheap.

Fig. 2.1 displays a sketch of the validity span for the different approaches to chemical modeling discussed in this section.

2.5

Computational approaches for reactive

Navier-Stokes equations

Finding analytical solutions to the Navier-Stokes equations (2.7-2.10) is, in most cases, impossible. Discretizing the different operators on a computational grid remains the main, if not the only, way to deal with this set of equations in three-dimensional configurations. The main difficulty with the numerical approach is the grid resolution needed to accu-rately solve the Navier-Stokes equations. This resolution depends on the range of time and length scales of the problem considered. An important issue when dealing with tur-bulent reactive flows, like explosions for example, is that this range can be quite large and is dictated by the coupling of turbulence and combustion processes. Even without combustion, turbulence itself is one of the most complex phenomenon in fluid mechanics. Indeed, turbulence can be characterized by fluctuations f0 of local properties f . The latter can then be split into mean f and fluctuating contributions: f = f + f0. The turbulence fluctuations are associated with different scales ranging from the largest one, the integral length scale lt, to the smallest one, the Kolmogorov scale ηk. lt is usually

close to the characteristic size of the geometry and defines the largest structures of the flow dominantly controlled by inertia. For structures smaller than lt, the viscous forces

become more and more important until they balance inertia at the Kolmogorov scale ηk.

Numerical simulations of turbulent reactive flows may be achieved using three levels of resolution:

• When the grid resolution is sufficient to capture all the length scales of the flow, no model is needed and the approach is called DNS for Direct Numerical Simulation. DNS is very costly due to the number of points needed to capture both flow and flame structures. Even though the field of High Performance Computing (HPC) has

Figure 2.2: Turbulent energy spectrum as a function of wave numbers. (a) RANS, LES and

DNS are conceptualized in terms of spatial frequency range. klt, kcand kη are the wave numbers

associated with the integral, cut-off and Kolmogorov length scales respectively.

known considerable break-through, DNS is still limited to canonical problems. DNS is however very useful to gain insight into complex mechanisms and the interplay among them, especially when a detailed description of the flow is needed. DNS will be used in this PhD to investigate the intricate DDT and inhibitor-flame interaction processes.

• On the other hand, when the averaged Navier-Stokes balance equations are solved, the approach reduces to RANS, for Reynolds Averaged Navier Stokes. The RANS method computes the mean quantities of interest, which requires closure rules: a tur-bulence model for the flow dynamics and a turbulent combustion model to describe heat release and chemical species conversion. RANS techniques are historically the first considered since solving the instantaneous turbulent flow was impossible. They are still the standard approach in all commercial codes for combustion today because of their low computational cost.

• In between RANS and DNS, there is LES for Large Eddy Simulation. This method consists in filtering the instantaneous balance equations using a cut-off length scale

lc, associated with wave numbers kc in Fig. 2.2. The large scales of turbulences,

namely larger than lc, are explicitly calculated whereas the effects of smaller ones,

namely smaller than lc, are modeled using subgrid closure rules. Because of its

reasonable cost compared to DNS, LES has become the standard research tool in the turbulent reactive flows community and is more and more used in the industry (during the motor design process for example). Note that LES is designed to tend toward DNS when the cut-off length scale, lc goes to zero. In this thesis, LES is

used to perform flame acceleration simulations. LES equations and closure models are described in details in the next section.

2.6

Large Eddy Simulations (LES)

LES methods consider a filtering operation performed on the Navier-Stokes balance equa-tions. Let the filter size be ∆. The filtering operation on a quantity f is given by:

f =

Z

f (x0)G∆(x − x0)dx0 (2.25)

where G∆is the filter kernel (low-pass filter). The contribution of the subscale structures,

i.e. structures smaller than the filter size ∆, is then given by: f0 = f − f . For compressible flows, a mass-weighted Favre Filtering is preferred:

ρf =e Z

ρf (x0)G∆(x − x0)dx0 (2.26)

At this point it is important to note that: 1) independently of the filtering operation, the filtered value of LES fluctuations f0 _{is not zero and filtering a filtered value f does not}

equal f ; 2) filtering the Navier-Stokes equations requires to commute integral and partial differential operators, which is not valid under most conditions. The error related to this exchange of operators is however neglected in general.

Filtered Navier-Stokes equations

Filtering the instantaneous balance equations leads to the following equations:

∂ρ ∂t + ∂ ∂xi (ρfui) = 0 (2.27) ∂ ∂t(ρfui) + ∂ ∂xi (ρfuiufj) = ∂ ∂xi (τi,j− ρ(ugiuj−fuiufj)) − ∂p ∂xi (2.28) ∂ ∂t(ρE) +e ∂ρfuiEe ∂xi = − ∂ ∂xi  ρ(ug_iE − f uiE)e  + ∂ ∂xi λ∂T ∂xi ) ! − ∂ ∂xi X k hs,kJi,k ! − ∂ ∂xi  ui(pδi,j − τi,j)  + ˙Q + ˙ωT (2.29) ∂ ∂t(ρYfk) + ∂ρYf_ku_fi ∂xi = − ∂ ∂xi  ρ(ug_iY_k−u_fiYf_k)  − ∂Ji,k ∂xi + ˙ωk (2.30)

Species and enthalpy laminar fluxes are generally negligible compared to turbulent fluxes. They can be either neglected or modeled using a simple gradient assumption:

Ji,k = −ρ Dk Wk W ∂gX_k ∂xi −Yf_kVf_ic ! (2.31) λ∂T ∂xi = λ∂Te ∂xi (2.32) X k hs,kJi,k = X k g hs,kJi,k (2.33) where Vf_ic =PN k  DkW_Wk∂_∂xXf_ik 

. In addition, transport properties are also estimated using resolved quantities: µ = µ(e T ), De k = µ/(ρSck) and finally λ = µCp(T )/P r. A numbere

of other terms in equations (2.27-2.30) can not be computed directly in LES and require closure models. This is discussed in the following.

Reynolds stress tensor Ti,j =ugiuj −fuifuj

This term is closed using a turbulence model. According to the Bousinesq assumption, unresolved momentum fluxes can be expressed as (Pope (2001)):

Ti,j = −µt ∂fui ∂xj +∂fuj ∂xi − 2 3δi,j ∂ufk ∂xk ! (2.34)

where µtis the subgrid scale viscosity. The different models for Ti,j differ in the expression

for µt. In this thesis, LES simulations were performed using the WALE model Nicoud

and Ducros (1999). This model has a number of advantages: • a correct asymptotic behavior at the walls;

• no subgrid viscosity is applied in the case of pure shear. The kinematic turbulent viscosity in the WALE model is given by:

νt = (Cw∆)2

(sd_i,jsd_i,j)3/2

(Sg_i,jSg_i,j)5/2+ (sd_i,jsd_i,j)5/4

(2.35) si,j = 1 2(ggi,j 2₊ g gj,i2) − 1 3ggk,k 2_δ i,j (2.36)

where Cw = 0.4929 is a constant of the model, ggj,i is the resolved velocity gradient tensor

and Sg_i,j is the symmetric part of the velocity gradient tensor.

Species and enthalpy turbulent fluxes

These therms are usually evaluated using resolved gradients:

ρ(ug_iY_k− f uiYf_k) = −ρ Dt k Wk W ∂Xg_k ∂xi −Yf_kVgc,t i ! (2.37)

where the species turbulent mass diffusion is given by Dt

k= µt/Sctk, Sctk being the species

turbulent Schmidt number, and _Vgc,t

i =

P

kDktWWk ∂Xfk

∂xi. In a similar manner, the following model is used for the turbulent enthalpy flux:

ρ(ug_iE − f uiE) = −λe t ∂Te ∂xi +X k g hs,kJi,k t (2.38)

where the turbulent heat diffusion coefficient is given by λt _{= ν}

tCp/P rt, P rt being the

turbulent Prandtl number.

Species chemical source term ˙ωk:

This term is crucial in reactive turbulent flows computations. Closure models for ˙ωk are

2.7

Numerical aspects

The AVBP solver

AVBP Gourdain et al. (2009) is a DNS/LES massively-parallel solver for the fully com-pressible reacting Navier-Stokes equations. The equations are solved explicitly on un-structured and hybrid grids. It relies on a cell-vertex discretization method and generally treats boundaries according to the Navier Stokes Characteristic Boundary Conditions (NSCBC) formalism Poinsot and Lele (1992).

Numerical schemes

Different numerical schemes can be used to solve equations 2.27-2.30. Their dissipation and dispersion properties are of paramount importance to accurately reproduce flow and flame structures. The convective numerical schemes used during this thesis are:

• The Lax-Wendroff (LW) scheme (Lax and Wendroff (1960)) is a finite

volume scheme with an explicit single step time integration. It is second order accurate in time and space. It has a low computational cost.

• The Two-step Taylor Galerkin (TTGC) scheme (Colin and Rudgyard

(2000)) is a finite element scheme, with an explicit two-step integration in time. It

is third order accurate in space and time. TTGC has much better dispersion and dissipation properties than LW, which makes it well-suited for LES applications. It is however approximately 2.5 times more expensive than the LW scheme.

The diffusion operator is descretized using the finite element diffusion scheme 2∆ devel-oped by Colin (2000).

Artificial viscosity

The fact that AVBP uses centered numerical schemes leads to simulations that are prone to point-to-point oscillations (wiggles) close to regions of steep gradients. To attenuate these spurious oscillations, two artificial viscosities are employed in practice:

• A background dissipation term (4th _{order hyperviscosity) attenuates the amplitude}

of the wiggles.

• A 2nd _{order viscosity term introduces artificial dissipation which smooths local}

gra-dients. A sensor, detecting strong deviations of variables from linear behavior, is employed to apply the 2nd order viscosity term only in regions where the sensor is triggered avoiding global dissipation of the solution. The sensor used in this thesis is the one introduced in Colin(2000).

Handling shocks

To properly deal with the strong shocks involved in DDT, the Cook and Cabot (2004) sensor is used to add a spectral-like viscosity to stabilize numerical solutions and reduce oscillations near discontinuities. The Cook and Cabot (2004) method consists in intro-ducing an additional bulk viscosity in the viscous stress tensor (Eq. (2.11)):

τ_i,jCook = τi,j+ k0

∂uk ∂xk δi,j (2.39) = (k0− 2 3µ) ∂uk ∂xk δi,j+ µ ∂ui ∂xj +∂uj ∂xi ! (2.40) A spectral-like and grid dependent model for k0 _{is chosen:}

k0 = Crρ∆r+2_x |∇^r_{S| (2.41)}

where ∆xis the grid spacing and S is the strain rate tensor S = ((_∂x∂u_ji+ ∂uj

∂xi)/2)i,j=1,3. ∇ r_S

is the polyharmonic operator denoting a series of Laplacians. For example, it corresponds to the biharmonic operator ∇4_{S = ∇}2_(∇2_{S) for r = 4. The f operator denotes ae}

truncated Gaussian filter similar to the one introduced in Eq. 2.25. For unstructured grids, the only feasible option is to consider a single Laplacian, so that, in AVBP, r is set to 2 with the model constant C2 _{= 5.}

Theoretical concepts of premixed

flames

3.1

Introduction

The goal of this chapter is to provide some theoretical aspects of premixed flames. First, intrinsic properties of premixed flames are introduced. We then focus on the dynamics of premixed flame fronts in the laminar and turbulent regimes of propagation. Finally, several approaches to turbulent combustion modeling are discussed.

3.2

Laminar premixed flames

3.2.1

Unstretched premixed flames

The unstretched premixed flame is a canonical case to understand fundamental combus-tion processes. It corresponds to the situacombus-tion where fuel and oxidizer are mixed prior to combustion. Temperature, pressure and composition of the mixture define the initial conditions of such flames. To define the composition, the equivalence ratio is used:

φ = sYF YO = Y F YO  / Y F YO  st

where YF and YO are the fuel and oxidizer mass fractions in the mixture. s is the mass

stoichiometric ratio which reads:

s = _Y O YF  st = ν 0 OWO ν0 FWF

where ν_F0 and ν_O0 are the fuel and oxidizer stoichiometric molar coefficients of the global reaction.

Flame structure

The first attempt to describe the flame structure is attributed to Zeldovich and Frank-Kamenetskii (1938) and remains the basis of most asymptotic methods. Considering an axis positioned in the reference frame of a one-dimensional flame results in a steady state

version of the Navier-Stokes balance equations. A number of assumptions can be used to further simplify the equations:

• combustion proceeds through a single irreversible exothermic reaction controlled by an Arrhenius law with a large activation energy;

• one of the reactive species is in excess, so that its concentration varies little and its consumption does not affect the reaction rate. Only one species can therefore be considered. The reaction can then be described by the relation R → P + Q, in which the reactant R is transformed into products P with the release of chemical energy Q;

• species R and P have the same molecular weight W , constant heat capacity Cp,

molecular diffusion coefficient D and Lewis number Le = 1. These assumptions yield the following set of equations:

∂ρu ∂x = 0 or equivalently ρu = cst = ρusL (3.1) ρuCpsL ∂T ∂x = ∂ ∂x λ ∂T ∂x ! − Q ˙ωR (3.2) ρusL ∂YR ∂x = ∂ ∂x ρD ∂YR ∂x ! + ˙ωR (3.3)

The above assumptions may seem strong, but the resulting equations preserve the global features of flames: intense non-linear heat release, variable density and temperature. A schematic of the obtained flame structure is displayed in Fig. 3.1(left), where 3 regions can be distinguished:

• a preheat zone, where fresh gases are heated due to thermal fluxes;

• a reaction zone, where the rate of the reaction R → P + Q increases drastically. Reactant R is converted into products P leading to chemical energy release Q; • a post-flame zone, where the temperature reaches the adiabatic equilibrium value. These simple models R → P + Q provide a global description of the flame structure and are useful to study theoretically some global flame parameters.

Intrinsic flame properties

One-dimensional laminar premixed flames have a characteristic consumption speed, sL

(Eq.3.1). The set of equations Eqs.3.1-3.3 can be integrated asymptotically (i.e. for very large activation energies) to obtain an analytical expression for sL Zeldovich and

Frank-Kamenetskii (1938); Williams (1985);Ferziger and Echekki (1993). A simple scaling law for sL can be deduced from an asymptotic analysis of the governing equations and shows

that:

Figure 3.1: Premixed flame structure using: (left) the simplest chemical model (R → P + Q)

which mimics reactants R consumption to form products P while producing chemical energy

Q; (right) detailed chemistry where the whole set of bi-molecular reactions are considered and

where intermediate species I_k are considered.

A first estimation of the flame thickness can be deduced from sL via scaling laws:

δth ∝ Dth/sL ∝ (Dth/A)1/2 (3.5)

δth is called the diffusive thickness and can be evaluated without flame computation.

There are other ways to define a flame thickness, based on the flame profile. The thermal thickness δL, for example, can be obtained from the temperature profile as:

δL=

Tb− Tu

max(|∂T_∂x|)

where Tb and Tu are the burnt and fresh gas temperature respectively. Tb corresponds to

the final temperature of the gas after all chemical energy has been released. At constant pressure, the following expression for Tb is obtained:

Tb = Tu + QYFu/Cp

where Yu

F is the fuel mass fraction in the fresh gases.

Flame structure using detailed chemistry

Some phenomena can not be described by the thermal runaway of the simple model presented in the last section. Important notions, like ignition, require to take into account the detailed reaction scheme. Indeed, in real flames, reactants are not directly converted into products. Instead, intermediate species Ik(see Fig.3.1(right)), which are absent from

the initial composition of the mixture, are produced via usually endothermic initiation reactions and interact with the reactants via chain branching reactions. The intermediate species are finally transformed into stable products Pk by strongly exothermic